# Properties

 Label 2-2671-1.1-c1-0-86 Degree $2$ Conductor $2671$ Sign $-1$ Analytic cond. $21.3280$ Root an. cond. $4.61822$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Origins

## Dirichlet series

 L(s)  = 1 − 2.34·2-s − 1.17·3-s + 3.51·4-s + 0.325·5-s + 2.75·6-s − 3.11·7-s − 3.55·8-s − 1.62·9-s − 0.764·10-s + 0.901·11-s − 4.12·12-s + 0.278·13-s + 7.30·14-s − 0.382·15-s + 1.31·16-s + 4.78·17-s + 3.81·18-s − 6.54·19-s + 1.14·20-s + 3.65·21-s − 2.11·22-s + 0.624·23-s + 4.17·24-s − 4.89·25-s − 0.654·26-s + 5.42·27-s − 10.9·28-s + ⋯
 L(s)  = 1 − 1.66·2-s − 0.677·3-s + 1.75·4-s + 0.145·5-s + 1.12·6-s − 1.17·7-s − 1.25·8-s − 0.540·9-s − 0.241·10-s + 0.271·11-s − 1.19·12-s + 0.0773·13-s + 1.95·14-s − 0.0986·15-s + 0.329·16-s + 1.15·17-s + 0.898·18-s − 1.50·19-s + 0.255·20-s + 0.796·21-s − 0.451·22-s + 0.130·23-s + 0.851·24-s − 0.978·25-s − 0.128·26-s + 1.04·27-s − 2.06·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2671$$ Sign: $-1$ Analytic conductor: $$21.3280$$ Root analytic conductor: $$4.61822$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{2671} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 2671,\ (\ :1/2),\ -1)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2671 $$1 + T$$
good2 $$1 + 2.34T + 2T^{2}$$
3 $$1 + 1.17T + 3T^{2}$$
5 $$1 - 0.325T + 5T^{2}$$
7 $$1 + 3.11T + 7T^{2}$$
11 $$1 - 0.901T + 11T^{2}$$
13 $$1 - 0.278T + 13T^{2}$$
17 $$1 - 4.78T + 17T^{2}$$
19 $$1 + 6.54T + 19T^{2}$$
23 $$1 - 0.624T + 23T^{2}$$
29 $$1 - 4.53T + 29T^{2}$$
31 $$1 - 10.5T + 31T^{2}$$
37 $$1 - 2.02T + 37T^{2}$$
41 $$1 + 5.23T + 41T^{2}$$
43 $$1 - 7.56T + 43T^{2}$$
47 $$1 + 10.7T + 47T^{2}$$
53 $$1 - 1.60T + 53T^{2}$$
59 $$1 - 8.21T + 59T^{2}$$
61 $$1 + 12.1T + 61T^{2}$$
67 $$1 - 11.3T + 67T^{2}$$
71 $$1 + 3.12T + 71T^{2}$$
73 $$1 + 2.36T + 73T^{2}$$
79 $$1 - 13.5T + 79T^{2}$$
83 $$1 + 12.4T + 83T^{2}$$
89 $$1 + 4.77T + 89T^{2}$$
97 $$1 - 13.8T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$